Receiver for and method of receiving symbols over time varying channels with doppler spread

This application is the U.S. National Phase Application of PCT International Application No. PCT/EP2022/079857, filed Oct. 26, 2022, which claims priority to German Application No. 10 2022 106 409.3, filed Mar. 18, 2022 and German Application No. 10 2021 212 161.6, filed Oct. 27, 2021, the contents of such applications being incorporated by reference herein.

The present invention relates to a method of receiving symbols over an orthogonal time frequency space (OTFS) communication channel subject to Doppler spread and a receiver implementing the method.

The sixth generation (6G) wireless communications and beyond are expected to serve a large number of high-mobility users, e.g., vehicles, subways, highways, trains, drones, low earth orbit (LEO) satellites, etc.

The preceding fourth and fifth generation (5G) wireless communications use orthogonal frequency division multiplexing (OFDM), which provides high spectral efficiency and high robustness against frequency selective fading channel, and also allow for using low-complexity equalisers. However, due to speed-dependent Doppler shifts or spreads and quickly varying multipath reception, high-mobility communications suffer from severe time and frequency dispersiveness. Time and frequency dispersiveness each cause signal fading at the receiver, and the fading is thus also referred to as doubly selective channel fading. Doubly selective channel fading significantly impairs the performance of OFDM communication.

As an alternative to OFDM, OTFS modulation was proposed as a solution for coping with doubly selective fading channels.

OTFS modulation is a 2D modulation scheme that multiplexes information QAM symbols over carrier waveforms that correspond to localized pulses in a signal representation that is referred to as delay-Doppler representation. The OTFS waveforms are spread over both time and frequency while remaining roughly orthogonal to each other under general delay-Doppler channel impairments. In theory, OTFS combines the reliability and robustness of spread spectrum with the high spectral efficiency and low complexity of narrowband transmission.

The OTFS waveforms couple with the wireless channel in a way that directly captures the underlying physics, yielding a high-resolution delay-Doppler Radar image of the constituent reflectors. As a result, the time-frequency selective channel is converted into an invariant, separable and orthogonal interaction, where all received symbols experience the same localized impairment and all the delay-Doppler diversity branches are coherently combined.

This renders OTFS ideally suited for wireless communication between transmitters and receivers that move at high speeds with respect to each other, e.g., receivers or transmitters located in high-speed trains, cars and even aircrafts.

However, OTFS presents its own challenges when it comes to channel estimation and equalisation in a receiver, and using adapted conventional OFDM receiver designs does not provide the required performance, requires significant pilot overhead of up to 50%, or provides acceptable performance only under ideal conditions, which are unrealistic in practice.

Throughout this specification, bold symbols represent vectors or matrices. Superscripts T, H and †, respectively denote the transpose, complex conjugate transpose and pseudo inverse of a vector or matrix. diag {a} is a diagonal matrix with vector a on its diagonal, while diag {A} is a vector whose elements are from the diagonal of matrix A. ⊗ is the Kronecker product.

An aspect of the present invention includes proposing a receiver for an OTFS transmission system and a corresponding method for receiving binary data sequences over an OTFS communication channel, in particular in OTFS communication channels having long delay spread and large Doppler spread, the receiver and method permitting using communication frames having a small pilot overhead or requiring no dedicated pilot slots at all while providing near-optimal performance from transmission to decoding.

The various aspects of the present invention rely on a novel model representing an OTFS channel, which will be introduced prior to discussing the application thereof in the novel receiver and the corresponding method for receiving.

shows a block diagram of a general OTFS transmission system. A transmittercomprises a first transmitter-side transformation unitand a second transmitter-side transformation unit. Serial binary data is input to a signal mapper (not shown in the figure) that outputs a two-dimensional sequence of information symbols x[k, l] in which the QAM symbols are arranged along the delay period and the Doppler period of the delay-Doppler domain. The information symbols comprise data symbols, pilot symbols and guard symbols surrounding the pilot symbols. The two-dimensional sequence of information symbols x[k, l] is input to the first transmitter-side transformation unitand is subjected to an inverse Finite Symplectic Fourier Transformation (ISFFT), which produces a matrix X[n, m] that represents the two-dimensional sequence of information symbols x[k, l] in the time-frequency domain. As the transmitter transmits in the time domain, a further transformation in the second transmitter-side transformation unitis required, which produces the signal s[t] in the time domain, e.g., a Heisenberg transformation. The signal s[t] is then transmitted via an antennaover the communication channel.

In a realistic environment the transmitted signal, on its way from the transmitter through the communication channel to the receiver, is subject to doubly selective fading with Doppler spread. The received signal is a superposition of a direct copy and a plurality of reflected copies of the transmitted signal, where each copy is delayed by a path delay that is dependent from the length of the signal's path delay and is frequency shifted by the Doppler shift that depends from the differential speed between transmitter, reflector, and receiver. Each of the signal copies is weighted in accordance with its particular path delay and differential speed. Typical Doppler shifts are on the order of 10 Hz-1 kHz, though larger values may occur in scenarios with extremely high mobility (e.g., high-speed trains) and/or high carrier frequency. As in realistic environments it is very likely that multiple reflectors and/or moving reflectors are present, the received superimposed signal is spread out over a frequency range rather than merely shifted in frequency, and the signal deformation is thus also referred to as Doppler spread. In the following description the realistic communication channel is also referred to as practical communication channel.

Inthe practical communication channel is represented by the undisturbed radio waves emitted from the transmitter antennaand the various unordered radio waves coming from different directions and with different distances to each other at the receiver antenna. The radio waves may arrive at the receiver's antenna directly or after being reflected one or several times at one or more stationary and/or moving objects, which may introduce Doppler shift and different delays to the reflected radio waves.

The receiverpicks up the received signal r[t] in the time domain, which is provided to a first receiver-side transformation unit, in which it is subjected to a Wigner transform for transforming the received signal r[t] into a matrix Y[n, m] representing the received signal r[t] in the time-frequency domain. For enabling signal detection in the delay-Doppler domain the matrix Y[n, m] is then provided to a second receiver-side transformation unit, where it is subjected to a Finite Symplectic Fourier Transformation (SFFT), which outputs a two-dimensional sequence of information symbols y[k, l] in the delay-Doppler domain. The two-dimensional sequence of information symbols y[k, l] is input to a channel estimation and equalisation block, which performs channel estimation CE and signal detection SD and reconstructs the symbols that were originally transmitted, and ultimately to a de-mapper that outputs the binary data that was originally transmitted (de-mapper not shown in the figure).

In order to enable channel estimation in the receiver, pilots may be added at the transmitter. These pilots, that are known beforehand at the receiver, are located at known positions within the two-dimensional sequence of information symbols that is ultimately transmitted. However, the pilots taking the place of data symbols, but not carrying any data, reduce the spectral efficiency of the system. In known OTFS receivers using CE-BEM channel estimation the pilot overhead, in order to achieve acceptable performance, must be increased with increasing maximum channel delay and Doppler spread, further reducing the spectral efficiency. While many OTFS channels may have a known maximum channel delay and possibly also maximum known Doppler spread, real-life systems will be designed for even higher maximum delay and Doppler spread, for providing some safety margin. This will even further reduce the spectral efficiency in such practical systems.

An improvement of the spectral efficiency can be achieved by using superimposed pilots and using the freed-up space for data symbols. Superimposed pilots employ low-powered pilots that are superimposed on data symbols in the delay-Doppler domain.

shows an illustration of superimposed pilots. As is shown in the left part of, the pilots may be arranged across the entire plane of the two-dimensional sequence of information symbols that are arranged along the delay period and the Doppler period of the delay-Doppler domain, albeit at a much lower power. The pilots are represented by the ordered checkerboard pattern, indicating the fact that the pilots are known beforehand at the receiver. The data is represented by the random pattern, indicating the variable nature of the data that is transmitted. The power allocation is indicated by the distance from the delay-Doppler plane. The right part ofshows an exemplary power allocation to pilots and data symbols. It is easy to see that the pilots have a much lower power than the data.

The data symbols and the pilots superimposed thereon are transformed into the OTFS signal vector, that is ultimately transmitted after further transformations.

In the following discussion of the transmitted signal M and N represent the dimensions of the delay grid and the Doppler grid, respectively, in which the symbols are arranged. The transmitted complex OTFS vector x, which consists of both superimposed pilots and data symbols, is defined as

In realistic scenarios, there is a constraint for the transmission power which covers both data and pilot transmission, i.e., data symbols and pilots share the total transmission power available to the transmitter. The transmitted complex OTFS vector x can be represented as a superimposed pilot vector xand a data vector x, in the delay-Doppler domain, which are defined as

Define Pas the total transmission power and α (α∈(0, 1)) as the pilot power allocation ratio. It suggests that αPand (1−α)Pare used for transmitting pilots and data symbols, respectively. As a result, the transmitted OTFS signal vector x can be expressed as

where α is the pilot power allocation ratio. Typically, if more power is used for pilot transmission, i.e., α is large, the channel estimation performance can be expected to be better. However, less power would remain for data transmission, giving rise to low data signal-to-noise-ratio (SNR) and thus low reliability. Instead, the pilots allocated with less power, i.e., α is small, would lead to a poor channel estimate and signal estimate. Therefore, a suitable power allocation between data and pilots is of utter importance in achieving high reliability.

The received OTFS vector y in the delay-Doppler domain is defined as

After propagating through the doubly-selective fading channel with Doppler spread, the received signal vector y, which can be considered the sum of the vectors representing the received data and the pilots superimposed thereon, respectively, in the delay-Doppler domain could be written as

where Fis the discrete Fourier transform (DFT) matrix, Ithe M×M identity matrix, w the additive white Gaussian noise (AWGN) vector, and Hthe MN×MN time varying channel matrix in the time domain defined as,

with h[t, l] denoting the channel gain of the l-th path at the t-th time instant, t=0, 1, . . . , MN−1, and l=0, 1, . . . , L. L denotes the channel length. Define

as the maximum Doppler frequency, where fis the carrier frequency, v the vehicle speed, and c the speed of light. Considering Jakes' model with U-shaped Doppler spectrum, the correlation function of the l-th path is defined as J(2πnfT), where J(⋅) denotes the zeroth-order Bessel function of the first kind, and Tthe sampling period.

The use of the delay-Doppler channel representation is beneficial due to its compactness and sparsity. Since typically there is only a small number of physical reflectors with associated reflected signals, far fewer parameters are needed for channel modelling and estimation in the delay-Doppler domain than in the time-frequency domain.

Some known OTFS receivers make use of the properties of the delay-Doppler channel representation and apply a basis expansion model (BEM) for parameterizing the time varying channel as a weighted combination of a number of basis functions in OTFS, making use of the fact that BEM can help reducing the number of unknown channel coefficients to be estimated, as will be shown further down.

BEM has numerous kinds, including complex exponential BEM (CE-BEM), generalized CE-BEM (GCE-BEM), non-critically sampled CE-BEM (NCS-CE-BEM), polynomial BEM, discrete prolate spheroidal (DPS) BEM, Karhunen-Loeve BEM (KL-BEM), spatial-temporal BEM, etc.

Among them, CE-BEM is the simplest model which, however, suffers a significant modelling error. On the positive side, CE-BEM and its variants GCE-BEM and NCS-CE-BEM are independent on the channel statistics. GCE-BEM enjoys simplicity and analytical tractability. However, its BEM order should be doubled at least, i.e., T≥2, for approaching near-optimal performance, where T is the modelling resolution parameter. Specifically, the GCE-BEM with T=1 suffers from a rather big modelling error, while the counterpart with T>1 enjoys low modelling error, albeit at the expense of large BEM order and high complexity.

In OTFS receivers using variants of the CE-BEM channel estimation, the pilot overhead, in order to achieve acceptable performance, must be increased with increasing maximum channel delay and Doppler spread. Such increase will evidently further reduce the spectral efficiency. While many OTFS channels may have a known maximum channel delay and possibly also maximum known Doppler spread, real-life systems will be designed for even higher maximum delay and Doppler spread, for providing some safety margin. This will even further reduce the spectral efficiency in such practical systems.

KL-BEM with a good knowledge of channel statistics is the most accurate BEM model. However, its performance is suboptimal when the assumed channel properties differ from the real channel.

The solution proposed herein, using superimposed pilots for an initial channel estimation and detected symbols as additional pseudo pilots in repeated iterative channel estimations, provides a way to benefit from the accuracy of the KL-BEM approach.

An important step is determining the most suitable basis functions for the KL-BEM channel estimation.

By exploiting the KL-BEM, Hcan be further expressed as

where Q is the BEM order, i.e., the number of BEM basis functions, which is usually given by

and E is the channel modelling error matrix. band Care defined as the q-th BEM basis function and its corresponding BEM coefficient, with q=0, 1, . . . , Q−1. Circulant matrix Ccan be expressed as

Thanks to the discrete Fourier Transform (DFT) concept, the circulant matric Ccan be represented by

where C=[c[0], c[1], . . . , c[L],]is the q-th BEM coefficient vector, Fis the MN-point DFT matrix, and Fcorresponds to the first (L+1) columns of F. As a result, H, can be expressed by

Thus, by applying BEM to OTFS, it is possible to reduce the number of unknown time varying channel coefficients from MN(L+1) to Q(L+1).

In the following, the optimal BEM basis functions will be investigated by taking the l-th channel path as an example. Define hand ĥas the channel and channel estimate of the l-th path, respectively. The mean square error (MSE) of the channel estimation is defined by

which is equivalent to